2003 cartier, makes a humming noise intermittently while driving, but most concerning is the same humming noise which happens even after the engine has been turned off, sometimes hours after it is turned off.
the other times it seems to happen is after breaking or when other changes are made in driving
Could you please elaborate on that concept?
I know what “breaking” is (and also “braking”), but I am at a loss to understand what “other changes in driving” might mean.
Additionally, if you could localize the origin of this humming noise, it would help us to help you.
Is that noise coming from…under the hood…behind the dashboard…underneath the car?
The humming you hear might be from the air compressor for the rear air-ride suspension.
Instead of having coil springs in the rear, your vehicle has air springs in the rear.
https://www.rockauto.com/en/moreinfo.php?pk=6370173&cc=1410714&jsn=416
If there’s a small leak in this system, the rear of the vehicle drops in level.
A sensor in the rear of the vehicle detects the drop in the rear of the vehicle, and turns the air compressor on to refill the springs with air until the rear of the vehicle is at the right level again.
Have the rear air-ride suspension system inspected for leaks.
Tester
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Seems to be under the hood, very loud. Other changes mean turning, turning on heater, etc but random and not consistent in any event.
Thanks. I’ll check it out. 
Open the gas cap and see if the humming stops. My car was doing that from the rear but you never know. That is just the easiest test.
Electric cooling fan maybe . I wouldn’t think it would run hours the engine was shut off though .
Sometimes the HVAC blend door actuator motors will make a noise with the engine off and key removed. Usually it’s described as a clicking or tapping rather than a humming. I think Tester’s idea is probably the problem, but if not keep the blend door actuators as a suspect too. Try pushing the heater and AC buttons when you hear this and the engine is running, see if that has any effect.